By a density and truncation argument
(as in [1]}),
we may assume without loss of generality that the fields \(u_\varepsilon \) 
are smooth and uniformly bounded.
Then, we apply [1]}.
For a given \(r > 0\)  (sufficiently small) and any \(\varepsilon \) ,
we find a finite collection of pairwise disjoint
balls \(B_{1,\varepsilon }, \, \ldots , \, B_{N_\varepsilon , \varepsilon }\) ,
of centers \(b_{k,\varepsilon }\)  and radii \(r_{k,\varepsilon }\) ,
that satisfy the following properties:
\(\sum _{k=1}^{N_\varepsilon } r_{k,\varepsilon } \le \frac{r}{16}  \\\left| u_\varepsilon (x) \right|_g \ge \frac{1}{2} \qquad \textrm {for } x\in M\setminus \bigcup _{k=1}^{N_\varepsilon } B_{k,\varepsilon }  \\\int _{B_{k,\varepsilon }}\left(\frac{1}{2}\left| D u_\varepsilon  \right|^2_g+ \frac{1}{4\varepsilon ^2}(1 - \left| u_\varepsilon  \right|_g^2)^2\right)\mathrm {vol}_g\ge \left| \operatorname{ind}(u_\varepsilon , \, \partial B_{k,\varepsilon }) \right|\left(\pi \log \frac{r}{\varepsilon } - C\right) \) 
for any \(k\) , \(\varepsilon \)  and some constant \(C\)  that depends only on \(M\) , \(C_0\) .
We claim that, for any \(\varepsilon \)  small enough and any \(k = 1, \, \ldots , \, N_\varepsilon \) ,
there holds
\( \operatorname{ind}(u_\varepsilon , \, \partial B_{k,\varepsilon })\ne 0\qquad \Longrightarrow \qquad b_{k,\varepsilon }\in \bigcup _{j=1}^n B_{r/2}(a_j)\) 
(We recall that the \(a_j\) 's are the points in the support
of the limit of \(\omega (u_\varepsilon )\) , i.e. \(\omega (u_\varepsilon ) \xrightarrow{}2\pi \sum _{j=1}^n d_j \delta _{a_j}\) .)
Suppose, towards a contradiction, that (REF )
is false. Then, up to a permutation of the indices \(k\) 
and extraction of a subsequence \(\varepsilon \rightarrow 0\) , we may assume that
\(\operatorname{ind}(u_\varepsilon , \, \partial B_{1,\varepsilon })\ne 0, \qquad b_{1,\varepsilon }\in M\setminus \bigcup _{j=1}^n B_{r/2}(a_j)\) 
By Equation (REF ), we obtain
\(B_{1,\varepsilon }\subseteq M\setminus \bigcup _{j=1}^n B_{r/4}(a_j)\) 
and hence, thanks to (),
\( \begin{split}\int _{M\setminus \bigcup _{j=1}^n B_{r/4}(a_j)}\left(\frac{1}{2}\left| D u_\varepsilon  \right|^2_g+ \frac{1}{4\varepsilon ^2}(1 - \left| u_\varepsilon  \right|_g^2)^2\right)\mathrm {vol}_g\ge \pi \log \frac{r}{\varepsilon } - C\end{split}\) 
However, the assumption () and
the lower energy bound ()
from Proposition REF , combined, give
\( \begin{split}\int _{M\setminus \bigcup _{j=1}^n B_{r/4}(a_j)}\left(\frac{1}{2}\left| D u_\varepsilon  \right|^2_g+ \frac{1}{4\varepsilon ^2}(1 - \left| u_\varepsilon  \right|_g^2)^2\right)\mathrm {vol}_g= \mathrm {o}_{\varepsilon \rightarrow 0}(\left| \log \varepsilon  \right|)\end{split}\) 
Equation (REF ) contradicts (REF ), and (REF ) follows.
Next, we claim that, for any \(\varepsilon \)  small enough and any \(j=1, \, \ldots , \, n\) ,
there exists \(k_* = k_*(j, \, \varepsilon ) \in \lbrace 1, \, \ldots , \, N_\varepsilon \rbrace \)  such that
\( \operatorname{ind}(u_\varepsilon , \, \partial B_{k_*,\varepsilon })\ne 0,\qquad b_{k_*,\varepsilon }\in B_{r/2}(a_j)\) 
Suppose, again towards a contradiction, that (REF )
does not hold. Then, there exists an index \(j\in \lbrace 1, \, \ldots , \, n\rbrace \) 
and a countable subsequence \(\varepsilon _m\rightarrow 0\) 
such that, for any \(m\)  and any \(k\in \lbrace 1, \, \ldots , \, N_{\varepsilon _m}\rbrace \) ,
\( b_{k,\varepsilon _m}\in B_{r/2}(a_j) \qquad \Longrightarrow \qquad \operatorname{ind}(u_{\varepsilon _m}, \, \partial B_{k,\varepsilon _m}) = 0\) 
By Fatou's lemma, we have
\(\begin{split}&{H}^1\left(\bigcup _{M\in \mathbb {N}}\bigcap _{m\ge M}\left\lbrace \rho \in (r/4, \, r/2)\colon \partial B_\rho (a_j) \cap \bigcup _{k=1}^{N_{\varepsilon _m}} B_{k,\varepsilon _m} \ne \emptyset \right\rbrace \right) \\&\qquad \qquad \le \liminf _{m\rightarrow +\infty } {H}^1\left\lbrace \rho \in (r/4, \, r/2)\colon \partial B_\rho (a_j)\cap \bigcup _{k=1}^{N_{\varepsilon _m}} B_{k,\varepsilon _m} \ne \emptyset \right\rbrace \stackrel{(\ref {H1b1})}{\le } \frac{r}{8}\end{split}\) 
Therefore, there exists a radius \(\rho \in (r/4, \, r/2)\)  and a (non-relabelled)
subsequence \(\varepsilon _m\rightarrow 0\)  such that
\( \partial B_\rho (a_j)\cap \bigcup _{k=1}^{N_{\varepsilon _m}} B_{k,\varepsilon _m} = \emptyset \) 
By Fubini theorem, we may also assume that
\(\int _{\partial B_\rho (a_j)} \left(\frac{1}{2}\left| D u_{\varepsilon _m} \right|^2_g+ \frac{1}{4\varepsilon ^2}(1 - \left| u_{\varepsilon _m} \right|^2_g)^2\right) \mathrm {d}{H}^1\lesssim \frac{C\left| \log \varepsilon _m \right|}{\rho }\) 
From (REF ), (REF ) and (REF ), we deduce that
\(\left| u_{\varepsilon _m}(x) \right|_g\ge 1/2\)  for any \(x\in \partial B_\rho (a_j)\)  and that
\(\operatorname{ind}(u_{\varepsilon _m}, \, \partial B_\rho (a_j)) = 0\) 
However, Lemma REF  implies
\(\operatorname{ind}(u_{\varepsilon _m}, \, \partial B_\rho (a_j)) = d_j \ne 0\) 
for any \(m\)  large enough. This is a contradiction, so (REF ) follows.
We can now conclude the proof of the lemma.
Without loss of generality, by taking \(r\)  small enough, we may
suppose that the balls \(B_r(a_j)\)  are pairwise disjoint.
By Equation (REF ), for any \(\varepsilon \) 
small enough there exist at least \(n\)  values of \(k\)  such that
\(\operatorname{ind}(u_\varepsilon , \, \partial B_{k,\varepsilon })\ne 0\) . By (REF )
and (REF ), for these values of \(k\)  we have
\(B_{k,\varepsilon }\subseteq \bigcup _{j=1}^n B_r(a_j)\) .
Then, () implies
\( \int _{\bigcup _{j=1}^n B_r(a_j)}\left(\frac{1}{2}\left| D u_\varepsilon  \right|^2_g+ \frac{1}{4\varepsilon ^2}(1 - \left| u_\varepsilon  \right|_g^2)^2\right)\mathrm {vol}_g\ge n\left(\pi \log \frac{r}{\varepsilon } - C\right)\) 
Combining (REF ) with (), the lemma follows.
